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()

()

(c)

ab

P

=

a b

P

(

)

(d)

a

PQ

+=+

a

P Q

a

(

)

(e)

ab

+=+

PPP

a

b

Using the associative and commutative properties of the real numbers, these

properties are easily verified through direct computation.

The
magnitude
of an
n
-dimensional vector
V
is a scalar denoted by
V
and is

given by the formula

n

2

V

=

V

.

(2.6)

i

i

=

1

The magnitude of a vector is also sometimes called the
norm
or the
length
of a

vector. A vector having a magnitude of exactly one is said to have
unit length
, or

may simply be called a
unit vector
. When
V
represents a three-dimensional point

or direction, Equation (2.6) can be written as

2

2

2

V

=

VVV

+

+

.

(2.7)

x

y

z

A vector
V
having at least one nonzero component can be resized to unit

length through multiplication by 1
V
. This operation is called
normalization
and

is used often in 3D graphics. It should be noted that the term
to normalize
is in no

way related to the term
normal vector
, which refers to a vector that is perpen-

dicular to a surface at a particular point.

The magnitude function given in Equation (2.6) obeys the following rules.

Theorem 2.2.
Given any scalar
a
and any two vectors
P
and
Q
, the following

properties hold.

(a)

P

≥

0

(b)

P

=

if and only if

P

=

0, 0,

, 0

0

(c)
a

P

=

a

P

(d)

PQ

+≤ +

P Q

Proof.

(a) This follows from the fact that the radicand in Equation (2.6) is a sum of

squares, which cannot be less than zero.

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